Download Lecture 2c - Newton`s Laws & Applications

PHY093 – Lecture 2c
Newton’s Laws & Applications
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Chapter 4
Dynamics: Newton’s Laws
of Motion
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Units of Chapter 4
• Force
• Newton’s First Law of Motion
• Mass
• Newton’s Second Law of Motion
• Newton’s Third Law of Motion
• Weight—the Force of Gravity; and the
Normal Force
• Solving Problems with Newton’s Laws:
Free-Body Diagrams
• Problem Solving—A General Approach
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4-1 Force
A force is a push or pull. An object
at rest needs a force to get it
moving; a moving object needs a
force to change its velocity.
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4-1 Force
Force is a vector, having both
magnitude and direction. The
magnitude of a force can be
measured using a spring
scale.
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4-2 Newton’s First Law of Motion
It may seem as though it takes a force to keep
an object moving. Push your book across a
table—when you stop pushing, it stops moving.
But now, throw a ball across the room. The ball
keeps moving after you let it go, even though
you are not pushing it any more. Why?
It doesn’t take a force to keep an object moving
in a straight line—it takes a force to change its
motion. Your book stops because the force of
friction stops it.
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4-2 Newton’s First Law of Motion
This is Newton’s first law, which is often
called the law of inertia:
Every object continues in its state of rest, or of
uniform velocity in a straight line, as long as no net
force acts on it.
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4-2 Newton’s First Law of Motion
Conceptual Example 4-1: Newton’s first law.
A school bus comes to a sudden stop, and all
of the backpacks on the floor start to slide
forward. What force causes them to do that?
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4-2 Newton’s First Law of Motion
Inertial reference frames:
Newton’s first law does not hold in every
reference frame, such as a reference frame that
is accelerating or rotating.
An inertial reference frame is one in which
Newton’s first law is valid. This excludes
rotating and accelerating frames.
How can we tell if we are in an inertial
reference frame? By checking to see if
Newton’s first law holds!
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4-3 Mass
Mass is the measure of inertia of an object,
sometimes understood as the quantity of
matter in the object. In the SI system, mass is
measured in kilograms.
Mass is not weight.
Mass is a property of an object. Weight is the
force exerted on that object by gravity.
If you go to the Moon, whose gravitational
acceleration is about 1/6 g, you will weigh much
less. Your mass, however, will be the same.
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4-4 Newton’s Second Law of Motion
Newton’s second law is the relation between
acceleration and force. Acceleration is
proportional to force and inversely proportional
to mass.
It takes a force to change
either the direction or the
speed of an object. More
force means more
acceleration; the same
force exerted on a more
massive object will yield
less acceleration.
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4-4 Newton’s Second Law of Motion
Force is a vector, so
coordinate axis.
is true along each
The unit of force in the SI
system is the newton (N).
Note that the pound is a
unit of force, not of mass,
and can therefore be
equated to newtons but
not to kilograms.
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4-4 Newton’s Second Law of Motion
Example 4-2: Force to accelerate a fast car.
Estimate the net force needed to accelerate
(a) a 1000-kg car at ½ g; (b) a 200-g apple at
the same rate.
Example 4-3: Force to stop a car.
What average net force is required to bring a
1500-kg car to rest from a speed of 100 km/h
within a distance of 55 m?
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4-5 Newton’s Third Law of Motion
Any time a force is exerted on an object, that
force is caused by another object.
Newton’s third law:
Whenever one object exerts a force on a second object,
the second exerts an equal force in the opposite
direction on the first.
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4-5 Newton’s Third Law of Motion
A key to the correct
application of the third
law is that the forces
are exerted on different
objects. Make sure you
don’t use them as if
they were acting on the
same object.
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4-5 Newton’s Third Law of Motion
Rocket propulsion can also be explained using
Newton’s third law: hot gases from combustion
spew out of the tail of the rocket at high speeds.
The reaction force is what propels the rocket.
Note that the
rocket does not
need anything to
“push” against.
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4-5 Newton’s Third Law of Motion
Conceptual Example 4-4: What exerts the
force to move a car?
Response: A common answer is that the
engine makes the car move forward. But it is
not so simple. The engine makes the wheels
go around. But if the tires are on slick ice or
deep mud, they just spin. Friction is needed.
On firm ground, the tires push backward
against the ground because of friction. By
Newton’s third law, the ground pushes on the
tires in the opposite direction, accelerating
the car forward.
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4-5 Newton’s Third Law of Motion
Helpful notation: the first subscript is the object
that the force is being exerted on; the second is
the source.
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4-5 Newton’s Third Law of Motion
Conceptual Example 4-5: Third law clarification.
Michelangelo’s assistant has been assigned the task of
moving a block of marble using a sled. He says to his boss,
“When I exert a forward force on the sled, the sled exerts an
equal and opposite force backward. So how can I ever start it
moving? No matter how hard I pull, the backward reaction
force always equals my forward force, so the net force must
be zero. I’ll never be able to move this load.” Is he correct? 19
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4-6 Weight—the Force of Gravity;
and the Normal Force
Weight is the force exerted on an
object by gravity. Close to the surface
of the Earth, where the gravitational
force is nearly constant, the weight of
an object of mass m is:
where
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4-6 Weight—the Force of Gravity;
and the Normal Force
An object at rest must have no net force on it. If
it is sitting on a table, the force of gravity is still
there; what other force is there?
The force exerted perpendicular to a surface is
called the normal force. It is
exactly as large as needed
to balance the force from
the object. (If the required
force gets too big,
something breaks!)
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4-6 Weight—the Force of Gravity;
and the Normal Force
Example 4-6: Weight, normal force, and a box.
A friend has given you a special gift, a box of
mass 10.0 kg with a mystery surprise inside.
The box is resting on the smooth
(frictionless) horizontal surface of a table.
(a) Determine the weight of the box and the
normal force exerted on it by the table.
(b) Now your friend pushes down on the box
with a force of 40.0 N. Again determine the
normal force exerted on the box by the table.
(c) If your friend pulls upward on the box with a
force of 40.0 N, what now is the normal force
exerted on the box by the table?
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4-6 Weight—the Force of Gravity;
and the Normal Force
Example 4-7: Accelerating
the box.
What happens when a
person pulls upward on the
box in the previous
example with a force
greater than the box’s
weight, say 100.0 N?
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4-6 Weight—the Force of Gravity;
and the Normal Force
Example 4-8: Apparent weight loss.
A 65-kg woman descends in an
elevator that briefly accelerates at
0.20g downward. She stands on a
scale that reads in kg.
(a) During this acceleration, what is
her weight and what does the
scale read?
(b) What does the scale read when
the elevator descends at a
constant speed of 2.0 m/s?
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4-7 Solving Problems with Newton’s Laws:
Free-Body Diagrams
1. Draw a sketch.
2. For one object, draw a free-body
diagram, showing all the forces acting
on the object. Make the magnitudes
and directions as accurate as you
can. Label each force. If there are
multiple objects, draw a separate
diagram for each one.
3. Resolve vectors into components.
4. Apply Newton’s second law to each
component.
5. Solve.
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4-7 Solving Problems with Newton’s Laws:
Free-Body Diagrams
Conceptual Example 4-10: The hockey puck.
A hockey puck is sliding at constant velocity across
a flat horizontal ice surface that is assumed to be
frictionless. Which of these sketches is the correct
free-body diagram for this puck? What would your
answer be if the puck slowed down?
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4-7 Solving Problems with Newton’s Laws:
Free-Body Diagrams
Example 4-11: Pulling the mystery box.
Suppose a friend asks to examine the
10.0-kg box you were given previously,
hoping to guess what is inside; and
you respond, “Sure, pull the box over
to you.” She then pulls the box by the
attached cord along the smooth
surface of the table. The magnitude of
the force exerted by the person is FP =
40.0 N, and it is exerted at a 30.0°
angle as shown. Calculate
(a) the acceleration of the box, and
(b) the magnitude of the upward force FN
exerted by the table on the box.
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4-7 Solving Problems with Newton’s Laws:
Free-Body Diagrams
Example 4-12: Two boxes connected
by a cord.
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Two boxes, A and B, are connected
by a lightweight cord and are resting
on a smooth table. The boxes have
masses of 12.0 kg and 10.0 kg. A
horizontal force of 40.0 N is applied
to the 10.0-kg box. Find (a) the
acceleration of each box, and (b) the
tension in the cord connecting the
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boxes.
4-7 Solving Problems with Newton’s Laws:
Free-Body Diagrams
Example 4-13: Elevator and
counterweight (Atwood’s machine).
A system of two objects suspended over
a pulley by a flexible cable is sometimes
referred to as an Atwood’s machine.
Here, let the mass of the counterweight
be 1000 kg. Assume the mass of the
empty elevator is 850 kg, and its mass
when carrying four passengers is 1150
kg. For the latter case calculate (a) the
acceleration of the elevator and (b) the
tension in the cable.
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4-7 Solving Problems with Newton’s Laws:
Free-Body Diagrams
Conceptual Example 4-14:
The advantage of a pulley.
A mover is trying to lift a
piano (slowly) up to a
second-story apartment.
He is using a rope looped
over two pulleys as shown.
What force must he exert
on the rope to slowly lift
the piano’s 2000-N weight?
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4-7 Solving Problems with Newton’s Laws:
Free-Body Diagrams
Example 4-15: Accelerometer.
A small mass m hangs from a
thin string and can swing like
a pendulum. You attach it
above the window of your car
as shown. What angle does
the string make (a) when the
car accelerates at a constant
a = 1.20 m/s2, and (b) when
the car moves at constant
velocity, v = 90 km/h?
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4-7 Solving Problems with Newton’s Laws:
Free-Body Diagrams
Example 4-16: Box slides
down an incline.
A box of mass m is placed on
a smooth incline that makes
an angle θ with the horizontal.
(a) Determine the normal
force on the box. (b)
Determine the box’s
acceleration. (c) Evaluate for a
mass m = 10 kg and an incline
of θ = 30°.
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4-8 Problem Solving—A General Approach
1. Read the problem carefully; then read it again.
2. Draw a sketch, and then a free-body diagram.
3. Choose a convenient coordinate system.
4. List the known and unknown quantities; find
relationships between the knowns and the
unknowns.
5. Estimate the answer.
6. Solve the problem without putting in any numbers
(algebraically); once you are satisfied, put the
numbers in.
7. Keep track of dimensions.
8. Make sure your answer is reasonable.
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Summary of Chapter 4
• Newton’s first law: If the net force on an object
is zero, it will remain either at rest or moving in a
straight line at constant speed.
• Newton’s second law:
• Newton’s third law:
• Weight is the gravitational force on an object.
• Free-body diagrams are essential for problemsolving. Do one object at a time, make sure you
have all the forces, pick a coordinate system and
find the force components, and apply Newton’s
second law along each axis.
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5-1 Applications of Newton’s Laws
Involving Friction
Friction is always present when two solid
surfaces slide along each other.
The microscopic details
are not yet fully
understood.
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5-1 Applications of Newton’s Laws
Involving Friction
Sliding friction is called kinetic friction.
Approximation of the frictional force:
Ffr = μkFN .
Here, FN is the normal force, and μk is the
coefficient of kinetic friction, which is
different for each pair of surfaces.
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5-1 Applications of Newton’s Laws
Involving Friction
Static friction applies when two surfaces
are at rest with respect to each other
(such as a book sitting on a table).
The static frictional force is as big as it
needs to be to prevent slipping, up to a
maximum value.
Ffr ≤ μsFN .
Usually it is easier to keep an object
sliding than it is to get it started.
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5-1 Applications of Newton’s Laws
Involving Friction
Note that, in general, μs > μk.
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5-1 Applications of Newton’s Laws
Involving Friction
Example 5-1: Friction: static and kinetic.
Our 10.0-kg mystery box rests on a horizontal floor. The
coefficient of static friction is 0.40 and the coefficient of kinetic
friction is 0.30. Determine the force of friction acting on the box
if a horizontal external applied force is exerted on it of
magnitude:
(a) 0, (b) 10 N, (c) 20 N, (d) 38 N, and (e) 40 N.
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5-1 Applications of Newton’s Laws
Involving Friction
Conceptual Example 5-2: A box against a wall.
You can hold a box against a rough
wall and prevent it from slipping
down by pressing hard horizontally.
How does the application of a
horizontal force keep an object
from moving vertically?
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5-1 Applications of Newton’s Laws
Involving Friction
Example 5-3: Pulling against friction.
A 10.0-kg box is pulled along a horizontal
surface by a force of 40.0 N applied at a 30.0°
angle above horizontal. The coefficient of
kinetic friction is 0.30. Calculate the
acceleration.
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5-1 Applications of Newton’s Laws
Involving Friction
Conceptual Example 5-4: To push or to pull a
sled?
Your little sister wants a
ride on her sled. If you
are on flat ground, will
you exert less force if
you push her or pull
her? Assume the same
angle θ in each case.
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5-1 Applications of Newton’s Laws
Involving Friction
Example 5-5: Two boxes and a pulley.
Two boxes are connected by a cord
running over a pulley. The coefficient of
kinetic friction between box A and the
table is 0.20. We ignore the mass of the
cord and pulley and any friction in the
pulley, which means we can assume
that a force applied to one end of the
cord will have the same magnitude at
the other end. We wish to find the
acceleration, a, of the system, which
will have the same magnitude for both
boxes assuming the cord doesn’t
stretch. As box B moves down, box A
moves to the right.
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5-1 Applications of Newton’s Laws
Involving Friction
Example 5-6: The skier.
This skier is descending a
30° slope, at constant
speed. What can you say
about the coefficient of
kinetic friction?
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5-1 Applications of Newton’s Laws
Involving Friction
Example 5-7: A ramp, a pulley, and two boxes.
Box A, of mass 10.0 kg, rests on a surface inclined at 37° to the
horizontal. It is connected by a lightweight cord, which passes
over a massless and frictionless pulley, to a second box B, which
hangs freely as shown. (a) If the coefficient of static friction is
0.40, determine what range of values for mass B will keep the
system at rest. (b) If the coefficient of kinetic friction is 0.30, and
mB = 10.0 kg, determine the acceleration of the system.
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